The spin stiffness at criticality - Boston University...

The spin stiffness at criticality

!s ! L!(d+z!2)

For a quantum-critical point with dynamic exponent z:

d=2, z=1 → plot Lρs vs g for different L• curves should cross (size independence) at gc

• x- and y-stiffness different in this model

Finite-size scaling in agreement with z=1, gc≈1.9094

28Friday, April 23, 2010

Valence-bond basis and resonating valence-bond statesAs an alternative to single-spin ↑ and ↓ states, we can use singlets and triplet pairs

static dimers(complete basis)

arbitrary singlets(overcomplete insinglet subspace)

one triplet in the “singlet soup” (overcompletein triplet subspace)

In the valence-bond basis (b,c) one normally includes pairs connectingtwo groups of spins - sublattices A and B (bipartite system, no frustration)

(a, b) = (!a"b # "a!b)/$

2 a ! A, b ! B

arrows indicate theorder of the spins in the singlet definition

Superpositions, “resonating valence-bond” (RVB) states

|!s! =!

!

f!|(a!1 , b!

1 ) · · · (a!N/2, b

!N/2)! =

!

!

f!|V!!

Coefficients fα>0 for bipartite (unfrustrated) Heisenberg systems• corresponds to Marshallʼs sign rule: sign=(-1)N↑(A), N↑(a) = # of spin-↑ on sublattice A


Calculating with valence-bond statesAll valence-bond basis states are non-orthogonal• the overlaps are obtained using transposition graphs (loops)

!V! | !V! |V"" |V!!

Many matrix elements can also be expressed using the loops, e.g.,

!V! |Si · Sj |V""!V! |V"" =

!0, for !i #= !j34"ij , for !i = !j

!i is the loop index (each loop has a label), staggered phase factor

!ij =!!1, for i, j on di!erent sublattices+1, for i, j on the same sublattice

More complicated cases derived in: K.S.D. Beach and A.W.S., Nucl. Phys. B 750, 142 (2006)

Each loop has two compatible spin states → !V! |V"" = 2Nloop!N/2

This replaces the standard overlap for an orthogonal basis; !!|"" = #!"


Solution of the frustrated chain at the Majumdar-Ghosh point

H =N!

i=1

"J1Si · Si+1 + J2Si · Si+2

#

We will show that these are eigenstates when J2/J1=1/2

Act with one “segment” of the terms of H on a VB state (J1=1, J2=g) :

Eigenstate for g=1/2; one can also show that itʼs the lowest eigenstate (more difficult)

H = !N!

i=1

(Ci,i+1 + gCi,i+2) + N1 + g

4,

Write H in terms of singlet projectors

Cij = !(Si · Sj ! 14 )

|!A! = |(1, 2)(3, 4)(5, 6) · · · !|!B! = |(N, 1)(2, 3)(4, 5) · · · !

Useful valence-bond results (easy to prove, just write as ↑ and ↓ spins)


2D Heisenberg results• h(r) = 1/r3

• good ground state properties• error in E < 0.1%, error in ms < 1%

Amplitude-product statesGood variational ground state for bipartite models can be constructed

|!s! =!

!

f!|(a!1 , b!

1 ) · · · (a!N/2, b

!N/2)! =

!

!

f!|V!!

Variational QMC methodGiven h(r), one can study the state using Monte Carlo sampling of bonds• elementary two-bond moves

• Metroplois accept/reject• loop updates when spins are included

• more efficient

f! =!

r

h(r)n!(r),

Let the wave-function coefficients be products of “amplitudes” (real positive numbers)

r is the bond length (n(r) = number of length r bonds)The amplitudes h(r) are adjustable parameters• use some optimization method to minimize E=<H>

Liang, Doucot, Anderson (PRL, 1990)


(-H)n projects out the ground state from an arbitrary state

Action of bond operators

H =!

!i,j"

!Si · !Sj = !!

!i,j"

Hij , Hij = (14 ! !Si · !Sj)

S=1/2 Heisenberg model

Project with string of bond operators

Hab|...(a, b)...(c, d)...! = |...(a, b)...(c, d)...!

Hbc|...(a, b)...(c, d)...! =12

|...(c, b)...(a, d)...!

!

{Hij}

n"

p=1

Hi(p)j(p)|!! " r|0! (r irrelevant)

Simple reconfiguration of bonds (or no change; diagonal)• no minus signs for A→B bond ‘direction’ convetion • sign problem does appear for frustrated systems

A BAB

(a,b)

(a,d)

(c,d)(c,b)

(i, j) = (| !i"j# $ | "i!j#)/%

2

Projector Monte Carlo in the valence-bond basisLiang, 1991; AWS, Phys. Rev. Lett 95, 207203 (2005)

(!H)n|!" = (!H)n!

i

ci|i" # c0(!E0)n|0"


Loop updates in the valence-bond basisAWS and H. G. Evertz, ArXiv:0807.0682

(ai, bi) = (!i"j # "i!j)/$

2

Put the spins back in a way compatible with the valence bonds

and sample in a combined space of spins and bonds

Loop updates similar to those in finite-T methods(world-line and stochastic series expansion methods)• good valence-bond trial wave functions can be used• larger systems accessible• sample spins, but measure using the valence bonds

|!!!!|

A

More efficient ground state QMC algorithm → larger lattices


J-Q model: T=0 results obtained with valence-bond QMCJ. Lou, A.W. Sandvik, N. Kawashima, PRB (2009)

Studies of J-Q2 model and J-Q3 model on L×L lattices with L up to 64

D2 = !D2x + D2

y", Dx =1N

N!

i=1

(#1)xiSi · Si+x, Dy =1N

N!

i=1

(#1)yiSi · Si+y

!M =1N

!

i

(!1)xi+yi !SiM2 = ! !M · !M"

Exponents ηs, ηd, and ν from the squared order parameters

Two different models: J-Q2 and J-Q3

Cij = 14 ! Si · Sj

H2 = !Q2

!

!ijkl"

CklCij

H3 = !Q3

!

!ijklmn"

CmnCklCij

H1 = !J!

!ij"

Cij

bond-singlet projector


Using coupling ratio

q =Qp

Qp + J, p = 2, 3

• AF order for q→0• VBS order for q→1

Finite-size scaling

J !Q2

J !Q3

J-Q2 model; qc=0.961(1)

!s = 0.35(2)!d = 0.20(2)" = 0.67(1)

J-Q3 model; qc=0.600(3)

!s = 0.33(2)!d = 0.20(2)" = 0.69(2)

Exponents universal (within error bars)• still higher accuracy desired (in progress)• there may be log-corrections (see arXiv:1001.4296)


Joint probability distribution P(Dx,Dy) of x and y columnarVBS order parameters

|0! =

!

k

ck|Vk!

QMC sampled state in thevalence-bond basis

Dx =!Vk| 1

N

!Ni=1("1)xiSi · Si+x|Vp#

!Vk|Vp#

Dy =!Vk| 1

N

!Ni=1("1)yiSi · Si+y|Vp#

!Vk|Vp#

Columnar or plaquette VBS?

4 peaks expected in VBS phase• Z4-symmetry unbroken in finite system

critical

Dx Dx

Dy Dy

columnar VBS plaquette VBS


VBS fluctuations in the theory of deconfined quantum-critical points➣ plaquette and columnar VBS “degenerate” at criticality➣ Z4 “lattice perturbation” irrelevant at critical point - and in the VBS phase for L<Λ∼ξa, a>1 ❨spinon confinement length❩➣ emergent U(1) symmetry➣ ring-shaped distribution expected for L<Λ

Dx Dx

Dy Dy

L=32J=0

AWS, Phys. Rev. Lett (2007)

No sign of cross-over to Z4 symmetricorder parameter seen in the J-Q2 model• length Λ > 32


Order parameter histograms P(Dx,Dy), J-Q3 model

This model has a more robust VBS phase• can the symmetry cross-over be detected?

D4 =!

rdr

!d!P (r,!) cos(4!)

VBS symmetry cross-over• Z4-sensitive order parameter

! ! !a ! q!a!

Finite-size scaling gives U(1)(deconfinement) length-scale

q = 0.635(qc ! 0.60)

L = 32

q = 0.85

L = 32

L1/a!(q ! qc)/qc

! ! 1.20± 0.05

J. Lou, A.W. Sandvik, N. Kawashima, PRB (2009)


The spin stiffness at criticality - Boston University...

Documents

Transcript of The spin stiffness at criticality - Boston University...